cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A124297 a(n) = 5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci(n).

Original entry on oeis.org

1, 11, 11, 31, 61, 151, 361, 911, 2311, 5951, 15401, 40051, 104401, 272611, 712531, 1863551, 4875781, 12760031, 33398201, 87424711, 228859951, 599129311, 1568486161, 4106261531, 10750188961, 28144128251, 73681909211, 192901135711
Offset: 0

Views

Author

Alexander Adamchuk, Oct 25 2006

Keywords

Comments

11 = Lucas(5) divides a(1+10k), a(2+10k), and a(9+10k). Last digit of a(n) is 1, or a(n) mod 10 = 1. For odd n there exists the so-called Aurifeuillian factorization A001946(n) = Lucas(5n) = Lucas(n)*A(n)*B(n) = A000032(n)*A124296(n)*A124297(n), where A(n) = A124296(n) = 5*F(n)^2 - 5*F(n) + 1 and B(n) = A124297(n) = 5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci(n).

Links

Crossrefs

Cf. A000032, A000045, A121171, A001946, A124296.
Bisections: A001604, A156095.

Programs

  • Mathematica
    Table[5*Fibonacci[n]^2+5*Fibonacci[n]+1,{n,0,50}]
LinearRecurrence[{4,-2,-6,4,2,-1},{1,11,11,31,61,151},30] (* Harvey P. Dale, Feb 23 2023 *)
  • PARI
    a(n)=subst(5*t*(t+1)+1,t,fibonacci(n)) \\ Charles R Greathouse IV, Jan 03 2013

Formula

a(n) = 5*Fibonacci(n)^2 + 5*Fibonacci(n) + 1.
G.f.: -(11*x^5-21*x^4-15*x^3+31*x^2-7*x-1) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). [Colin Barker, Jan 03 2013]

A156094 5 F(2n) (F(2n) - 1) + 1 where F(n) denotes the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 31, 281, 2101, 14851, 102961, 708761, 4865911, 33372361, 228792301, 1568309051, 10749725281, 73680695281, 505017569551, 3461448647801, 23725139605861, 162614572159411, 1114576979567761, 7639424583421961, 52361395886149351
Offset: 0

Views

Author

Stuart Clary, Feb 04 2009

Keywords

Comments

Natural bilateral extension (brackets mark index 0): ..., 15401, 2311, 361, 61, 11, [1], 1, 31, 281, 2101, 14851, ... This is A156095-reversed followed by A156094, without repeating the central 1. That is, A156094(-n) = A156095(n).

Crossrefs

Cf. A124296, A124297, A001603, A001604, A156095

Programs

  • Mathematica
    a[n_Integer] := 5 Fibonacci[2n] (Fibonacci[2n] - 1) + 1
5(#*(#-1))&/@Fibonacci[Range[0,40,2]]+1 (* Harvey P. Dale, Jan 06 2013 *)

Formula

Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
Alternate formula: a(n) = L(4n) - 5 F(2n) - 1.
Recurrence: a(n) - 10 a(n-1) + 23 a(n-2) - 10 a(n-3) + a(n-4) = -5.
Recurrence: a(n) - 11 a(n-1) + 33 a(n-2) - 33 a(n-3) + 11 a(n-4) - a(n-5) = 0.
G.f.: A(x) = (1 - 10 x + 53 x^2 - 60 x^3 + 11 x^4)/(1 - 11 x + 33 x^2 - 33 x^3 + 11 x^4 - x^5) = (1 - 10 x + 53 x^2 - 60 x^3 + 11 x^4)/((1 - x) (1 - 7 x + x^2) (1 - 3 x + x^2)).
a(n)=A056854(n)-5*A001906(n)-1. - R. J. Mathar, Feb 23 2009
a(n)=((2*sqrt(5))/2)*(((3-sqrt(5))/2)^n-((3+sqrt(5))/2)^n)+((7+3*sqrt(5))/2)^n+((7-3*sqrt(5))/2)^n-1. - Tim Monahan, Aug 15 2011

A156194 Period 12: 1,2,7,1,7,2,1,1,4,2,4,1 repeated.

Original entry on oeis.org

1, 2, 7, 1, 7, 2, 1, 1, 4, 2, 4, 1, 1, 2, 7, 1, 7, 2, 1, 1, 4, 2, 4, 1, 1, 2, 7, 1, 7, 2, 1, 1, 4, 2, 4, 1, 1, 2, 7, 1, 7, 2, 1, 1, 4, 2, 4, 1, 1, 2, 7, 1, 7, 2, 1, 1, 4, 2, 4, 1, 1, 2, 7, 1, 7, 2, 1, 1, 4, 2, 4, 1, 1, 2, 7, 1, 7, 2, 1, 1, 4, 2, 4, 1
Offset: 0

Views

Author

Paul Curtz, Feb 05 2009

Keywords

Comments

Also the decimal expansion of 42390704747/333333333333. - R. J. Mathar, Feb 23 2009

Links

Programs

  • Mathematica
    PadRight[{},144,{1,2,7,1,7,2,1,1,4,2,4,1}] (* Harvey P. Dale, Mar 06 2012 *)

Formula

Palindromic properties: a(12k+i) = a(12k+6-i), i=0..3. a(12k+7+i) = a(12k+11-i), i=0..2, and similarly for successive differences.
a(n) = A156095(n) mod 9.
a(n) = A156094(n+6) mod 9.
a(4n) + a(4n+1) + a(4n+2) + a(4n+3) = A010850(n).
G.f.: (1+2*x+7*x^2+x^3+7*x^4+2*x^5+x^6+x^7+4*x^8+2*x^9+4*x^10+x^11)/((1-x)*(1+x+x^2)*(1+x)*(1-x+x^2)*(1+x^2)*(x^4-x^2+1)). - R. J. Mathar, Feb 23 2009

Extensions

Edited by R. J. Mathar, Feb 23 2009
More terms from Jinyuan Wang, Feb 26 2020
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.